Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves∗

In this paper we prove that there is an infinite sequence of pairs of plane cuspidal curves, C_m,1 and C_m,2, of degree deg(C_m,1)=deg(C_m,2)→∞, such that the pairs $\text{(}\text{CP}{}^2\text{,C}{}_{\mathrm{m,1}}\text{)}$ and $\text{(}\text{CP}{}^2\text{,C}{}_{\mathrm{m,2}}\text{)}$ are diffeomorphic, but C_m,1 and C_m,2 have non-equivalent braid monodromy factorizations. These curves give rise to the negative solutions of “Dif ⇒ Def” and “Dif ⇒ Iso” problems for plane irreducible cuspidal curves. In our examples, C_m,1 and C_m,2 are complex conjugate.